Although the GLLMM framework does indeed treat random coefficient models (which include hierarchical and longitudinal models) as latent variable models, the framework leaves out important multivariate data reduction methods such PCA, Independent Component Analysis (ICA), Canonical Correlation Analysis (CCA) from the formulation. The argument put forward by [1]} contends that PCA and similar procedures are not latent variable models because they cannot be formulated as statistical models with latent variables corresponding to lower dimensional space. However, their assertion does not seem to have taken into consideration results that had been previously published in the literature. More specifically, in recent years, research in latent variable modeling has undergone a very active resurgence in the field of unsupervised machine learning. Indeed, great progress has been made in developing new latent variable models and recasting existing clustering and dimensionality reduction methods in a probabilistic graphical model formulation. For example, building on previous research in neural networks [2]}, [3]}, [4]} demonstrated that PCA can be indeed be formulated as a probabilistic latent variable model. In their paper, they showed that Probabilistic PCA is a special case of Factor Analysis (FA). More specifically, whereas in FA the error covariance matrix is diagonal, in Probabilistic PCA, it is \(\sigma \mathbf {I}\)  and standard PCA is obtained when \(\sigma \rightarrow 0\) . [5]} review several common methods showing that not just PCA and FA, but also ICA, Kalman Filters, and Hidden Markov Models (HMMs) can be considered as special cases of a general linear-Gaussian latent variable model. A probabilistic formulation of CCA [6]} later followed. Recently, a Group Factor Analysis (GFA) model has been proposed by [7]} that generalizes Probabilistic CCA to more than two data sets.
